THE 



PHILOSOPHICAL MAGAZINE 

 AND JOURNAL. 



3P' JANUARY 1825. 



I. On the Method of the Least Squares. By J. Ivory, Esq 

 M.A. F.E.S. 



T^HE present advanced state of astronomy is to be ascribed 

 chiefly to the precision of the theories, and to the great 

 exactness of modern observations. But there is another cir- 

 cumstance which has likewise contributed much to the same 

 end; and that is, the method of combining the observations 

 so as to draAV from them the most advantageous results. In 

 general a very few observations are sufficient for approxima- 

 ting to the quantity of an astronomical element with a consi- 

 derable degree of precision. But the errors unavoidable by 

 the most skilful observers must always leave some uncertainty 

 in such determinations. The utmost precision can only be 

 obtained by multiplying the observations, and by making a 

 great number, embracing every possible variety of circum- 

 stances, concur in the valuation of the same element. Astro- 

 nomers accomplish this end by the use of eqtiations of con- 

 dition. 



Let V denote an astronomical function, varying in different 

 circumstances, but involving one, or several determinate ele- 

 ments. Suppose that the values of the elements have been 

 found nearly ; and let the corrections, or the" differences be- 

 tween the approximate and the exact values, be denoted by 

 the small quantities x, ij, z, &c. : then if, for every element, we 

 substitute the sum of the approximate value and the correc- 

 tion, and expand into a series, neglecting the powers and 

 products of the small quantities x, jj, z, &c., we shall obtain, 



y = V' + ax + hi/ + cz + &c. 



L(!t O be an observed value of V: if () were rigorously 



exact, we should have O = V ; but as every observation is 



liable to error, if we denote tlie error by c, we shall have 



Vol. 65. No. :J21. Jati. 182.5. A 2 O I- 



